PHYSICAL REVIEW B 102, 245115 (2020) Memory-Critical Dynamical

PHYSICAL REVIEW B 102,245115(2020)

Memory-critical dynamical buildup of phonon-dressed Majorana fermions

Oliver Kaestle ,1,* Yue Sun, 2,3 Ying Hu,2,3 and Alexander Carmele 1 1Institut für Theoretische Physik, Nichtlineare Optik und Quantenelektronik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany 2State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University, Taiyuan, Shanxi 030006, China 3Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China

(Received 2 July 2020; revised 24 November 2020; accepted 24 November 2020; published 11 December 2020)

We investigate the dynamical interplay between the topological state of matter and a non-Markovian dissipation, which gives rise to a crucial timescale into the system dynamics due to its quantum memory. We speciﬁcally study a one-dimensional polaronic topological superconductor with phonon-dressed p-wave pairing, when a fast temperature increase in surrounding phonons induces an open-system dynamics. We show that when the memory depth increases, the Majorana edge dynamics transits from relaxing monotonically to a plateau of substantial value into a collapse-and-buildup behavior, even when the polaron Hamiltonian is close to the topological phase boundary. Above a critical memory depth, the system can approach a new dressed state of the topological superconductor in dynamical equilibrium with phonons, with nearly full buildup of the Majorana correlation.

DOI: 10.1103/PhysRevB.102.245115

I. INTRODUCTION phonon-renormalized Hamiltonian parameters (see Fig. 1). In contrast to Markovian decoherence that typically destroys Exploring topological properties out of equilibrium is cen- topological features for long times, we show that a ﬁnite tral in the effort to realize, probe, and exploit topological states quantum memory allows for substantial preservation of topo- of matter in the laboratory [1–15]. A paradigmatic scenario logical properties far from equilibrium, even when the polaron is where the topological system is coupled to a Markovian Hamiltonian is close to the topological phase boundary [see bath, inducing open-dissipative dynamics that is described by Fig. 2(b)]. Depending on the memory depth (i.e., the charac- aLindblad-formmasterequationforthetime-evolvedreduced teristic timescale of the quantum memory), the Majorana edge system density operator [16–28]. Yet solid-state realizations dynamics can monotonically relax to a plateau, or it undergoes of topological matter, such as topological superconductors acollapse-and-builduprelaxation[seeFig.2(c)]. Remarkably, [29–40], are often based on semiconductor nanostructures, when the memory depth increases above a critical value, the which are coupled to a structured phonon environment as edge correlation can nearly fully build up, corresponding to aresultofthedeformationofhostmaterialsanditslattice a new polaronic state of the topological superconductor in vibrations. In this case, the Markovian approximation and thus dynamical equilibrium with phonons. the Lindblad formalism usually fail. Compared to Markovian scenarios, key differences arise from the presence of quantum memory effects in non-Markovian processes: The information II. POLARONIC KITAEV CHAIN is lost from the system to the environment but ﬂows back at a later time [41,42]. This generates a timescale into the Concretely, we consider the paradigmatic Kitaev p-wave system dynamics that is strictly absent in a Markovian context superconductor [43], with super-Ohmic coupling to a three- and raises the challenge as to what the unique dynamical dimensional (3D) structured phonon reservoir. The total consequences of this interplay between the topological state Hamiltonian is denoted by H0 Hk Hb (¯h 1). The N 1= +† ≡ of matter and non-Markovian dissipation are. Kitaev Hamiltonian Hk l −1 [( Jcl cl 1 !cl cl 1) = = − + + + + Here, we demonstrate that the quantum memory from H.c.] µ N c†c describes spinless fermions c and c† on a l 1 l l ! l l anon-Markovianparity-preservinginteractionofatopo- chain− of N sites= l,withanearest-neighbortunnelingamplitude logical p-wave superconductor with surrounding phonons J R,pairingamplitude! ! R,andchemicalpotentialµ. can give rise to intriguing edge mode relaxation dynam- When∈ µ < 2J and ! 0, it∈ is in the topological regime fea- | | ̸= ics without a Markovian counterpart. Our study is based turing unpaired Majorana edge modes γL/R j 1 fL/R, jb j = = on the polaron master equation, describing open-dissipative [here, we use the Majorana operators b c c† and 2 j 1 ! j j dynamics of a polaronic topological superconductor with † − = + b2 j i(c j c j )], with fL/R, j being exponentially localized near= the− left− (L)andright(R)edges.TheMajoranamodes exhibit a nonlocal correlation θ i γLγR 1thatcor- *[email protected] responds to the fermionic parity= of− the⟨ (degenerate)⟩=± ground

Thus, the considered fermion-phonon interaction results in a polaronic Kitaev chain featuring phonon-dressed p-wave pairing, with phonon-induced quantum fluctuations. In deriving Eq. (1), we have ignored an arising energy renormalization 3 2 N † 2 term Hshift d k (gk/ωk )( l 1 cl cl ) ,knownasthepo- laron shift.= In the context of non-Markovian= dynamics, it FIG. 1. Left: A polaronic Kitaev chain, with phonon-dressed is standard practice" to neglect! this term if the initial condi- spinless fermions, exhibits a renormalized p-wave pairing B ! at tion is equilibrium [60,65], which is the case in the present temperature T [see Eq. (1)]. In the topological ground state,⟨ ⟩ two work, where the initial state is assumed to be the ground unpaired Majorana edge modes, γL and γR,emerge.Thecoupling state of an ideal Kitaev chain. However, to ensure validity gk to the structured phonon bath features mode dependence with of our results even in the presence of this energy renormal- spectral width σ . Right: Gaussian profile of gk in momentum space ization, in Appendix A we provide additional numerically for σ 0.2(green)andσ 0.6(red). = = exact calculations in the presence of the polaron energy shift, demonstrating that its inclusion does not affect the essential states. The entire chain is coupled to a 3D structured phonon physics presented below. Before tracing out the phononic degrees of freedom, we rewrite Eq. (1)asHp Hp,s Hp,I reservoir with a parity-preserving interaction, described by † = + + † N † † 3 3 Hp,b,withHp,b d k ωkrk rk for the reservoir. To recover Hb d k [ωkrk rk l 1 gkcl cl (rk rk )] [44–52]. Here, = = + = + the bare Kitaev Hamiltonian dynamics for the limiting case the operator r(†) annihilates (creates) phonons with momen- " k ! g 0, we introduce" a Franck-Condon renormalization of tum k and frequencies ω c k,wherec is the sound k k s s H →satisfying Tr [H ,ρ(t )] 0, with ρ(t )denotingthe velocity of the environment.= This coupling is commonly ap- p,I B p,I total density operator.{ The renormalized}= system Hamiltonian plied for the description of polaron formation [53–58]and H is given by in solid-state setups can represent the coupling of semicon- p,s ductor nanostructures to longitudinal acoustic phonons, which N 1 N − † † † † typically feature a long wavelength. We choose a generic Hp,s [ Jcl cl 1 ! B cl 1cl H.c.] µ cl cl , (2) = − + + ⟨ ⟩ + + − super-Ohmic fermion-phonon coupling gk which features fre- l 1 l 1 $= $= quency dependence modeled as a Gaussian function, gk where the pairing renormalization factor B is given explic- 2 2 2 = fph k/σ exp ( k /σ ), with σ being the width and fph be- ⟨ ⟩ − itly below. The system-reservoir interaction in the polaron pic- ing a dimensionless amplitude. We consider the topological N 1 2(R R† ) † † # ture reads Hp,I ! l −1 [(e− − B )cl 1cl H.c.]. superconductor and bath to be initially in equilibrium at low Wewillfocusonthelimit= = B 1,−⟨ which⟩ allows+ + us to treat temperatures, before a fast increase in the bath temperature ! ⟨ ⟩≪ Hp,I perturbatively in second-order Born theory, as dynamical induces an open-system dynamics. decoupling effects cannot occur [57,60]. In Appendix B,we provide a detailed derivation of the polaron master equation III. POLARON MASTER EQUATION for the reduced system density matrix ρS (t )ofthepolaron chain, obtaining At the heart of our following solution lies the polaron t representation of the coupled system (see Fig. 1). We de- 2 ∂t ρS (t ) i[Hp,s,ρS (t )] B dτ rive a master equation in second-order perturbation theory =− −⟨ ⟩ 0 of the dressed-state system-reservoir Hamiltonian [59]. In % ( cosh [φ(τ )] 1 [Xa, Xa( τ )ρS (t )] this dressed-state picture, the coherent process (i.e., higher- × { − } − sinh [φ(τ )][Xb, Xb( τ )ρS (t )] H.c.). (3) order contributions) from the fermion-phonon interaction is − − + retained through phonon-renormalized Hamiltonian parame- N 1 † † Here, Xa ! l −1 (cl cl 1 cl 1cl )andXb ters [53–58,60–64]. In this way, we efficiently account for the N 1 † †=− = + + + = ! − (c c c c )denotecollectivesystem non-Markovian character of the dynamics in the long-time l 1 l l 1 !l 1 l operators,= whose+ − dynamics+ obeys a time-reversed unitary limit not accessible and not captured in the typical second- evolution! governed by the renormalized Hamiltonian H , order Born approximation of the bare-interaction Hamiltonian p,s X ( τ ) e iHp,sτ X eiHp,sτ . φ(τ )representsthephonon H [44–47]. a,b − a,b b correlation− ≡ function, Defining collective bosonic operators R† d3k (g /ω )r†,weapplyapolarontransformation= k k k φ(τ ) d3k 2g (σ )/ω 2 N † † † (R R† ) † k k Up exp[ l 1 cl cl (R R)] resulting in cl e− − cl , = | | " = = − → % which describes phonon dressing of fermions. The h¯ωk transformed! total Hamiltonian H U H U 1 is derived coth cos (ωkτ ) i sin(ωkτ ) , (4) p p 0 p− × 2k T − as ≡ & ' B ( ) with kB being the Boltzmann constant. The renormalization N 1 − † 2(R R† ) † † factor B in Eq. (2)isdeterminedbytheinitialphononcor- Hp [ Jcl cl 1 !e− − cl 1cl H.c.] ⟨ ⟩ = − + + + + relation, B exp[ φ(0)/2], with temperature dependence l 1 ⟨ ⟩= − $= [66]. N Equation (3)providesthekeyequationforourstudy.Itfea- † 3 † µ c cl d k ωkr rk. (1) tures a phonon-renormalized Hamiltonian H and a memory − l + k p,s l 1 kernel involving both reservoir and system correlators φ(τ ) $= %

245115-2 MEMORY-CRITICAL DYNAMICAL BUILDUP OF … PHYSICAL REVIEW B 102,245115(2020)

(a) (b) (c) 10 1 1 Re[Φ(τ)] Im[Φ(τ)] Abs[Φ(τ)] 5 0.5 0.5 (t)

θ 1 (t) ) θ

0 0 ∞ σ = 0.4 (t 0 θ 0.2 σ = 0.2 0.2 σ 0.8 -5 -0.5 0 25 50 0 0.25 0.5 0 0.5 1 Jτ 100 Jt 100 Jt

FIG. 2. Non-Markovian dynamics of the polaronic topological superconductor. (a) Phonon correlation function φ(τ )forbandwidthσ = 0.2(green)andσ 0.6 (red) of the fermion-phonon coupling gk . (b) Comparisons of Majorana correlation θ(t )calculatedusingthetime- independent Lindblad-type= master equation for dephasing (solid black line) and Eq. (3) with fully accounting for memory (solid blue line). The dashed black line shows asymptotic θ(t ) in a coherent quench scenario ! ! B .(c)Non-Markoviandynamicsofθ(t ) for various bandwidths σ of phonon coupling. The corresponding steady-state value θ(t )isshownasafunctionof→ ⟨ ⟩ σ in the inset. In all plots, we choose ∞ 1 N 4, J ! 0.01, µ 0; the amplitude of g is taken to be f 0.1, and phonon modes within k [0.0, 4.0] nm− are considered. = = = = k ph = ∈ and Xa,b( τ ). Crucially, the correlation of a super-Ohmic correlation would approach an asymptotic value determined phonon bath− has a finite lifetime [see Fig. 2(a)], φ(τ ) 0 by the overlap of the edge mode wave functions for the pre- ≈ for τ>τM .Therefore,attimest >τM ,onlythoseXa,b( τ ) and postquench topological Hamiltonians [24], which is small − within the past times τ ! τM contribute to the integral in if the postquench Hamiltonian is near the phase boundary Eq. (3). That is, the memory has a finite depth τM ,sothat (dashed black line). This differs significantly from the non- ∼ for t >τM the integral becomes essentially time independent. Markovian behavior in Fig. 2(b) and underscores the essential The memory effect is substantial when Xa,b( τ )evolveson role of the memory effect. 1 − atimescale,typicallydeterminedbyE!− ,whereE! is the Auniquefeatureofthememoryeffectisthatit,be- bulk gap of Hp,s,comparabletoτM .Forgiventemperature cause of the dependence on both φ(τ )andthereversed and sound velocity cs, φ(τ )andτM critically depend on the dynamics of system correlations Xa,b( τ ), simultaneously − bandwidth σ of fermion-phonon coupling gk [see Fig. 2(a)]: introduces decoherence and backflow of coherence. The dy- Alargerσ yields a smaller τM and at the same time leads to a namical consequence of these two competing processes can smaller φ(0) and thus a larger B . be intuitively understood as follows: The dressed Kitaev Below we investigate the⟨ Majorana⟩ mode correlation wire initially in its ground state is perturbed by a tem- 2N θ(t ) iTr[ρS (t )γLγ j] i,l 1 fL,i fR,l *il (t )[24]. Here, perature increase to T ,resultinginarenormalizationof =− =− = * (t ) i Tr ρ(t )[b , b ] ,and f are the Majorana wave the polaron chain towards the phase boundary via φ(0) il 2 i l ! L/R,i 3 2 = functions= of{ the initial Kitaev} Hamiltonian in the Majorana d k 2gk (σ )/ωk coth [¯hωk/(2kBT )], which generates significant| bulk excitations| and populates the Majorana edge basis. For concreteness, we assume the system is initially in " the even-parity ground state of a perfect Kitaev chain with mode, changing the parity of Majorana states. Combined with phonon-assisted dephasing, this leads to strong decoherence ! J and µ 0, where fL,i δi,1 and fR,l δl,2N ,such = = = = in the polaronic Kitaev wire. On the other hand, the reversed that θ(t )reducestoθ(t ) *1,2N (t ). Then, a fast increase of =− dynamics of Xa,b( τ )actstoreinstatethecoherenceofthe temperature to T 4K resultsin ! ! B of the dressed − Hamiltonian, thus= inducing the dynamics→ of⟨ the⟩ polaron chain p-wave pairing that is the key ingredient for a topological for times t > 0. Due to the numerically very expensive size of wire. Such a rephasing effect is marginal, at times smaller than the density matrix and its memory kernel, computations are τM , so an irreversible loss of parity information dominates performed for N 4sites. the short-time dynamics. Once φ(τ )decaystozeroatlarge = times τ>τM ,thememoryreachesitsfulldepth,andthe rephasing of topological properties grows due to X ( τ ), IV. MEMORY: LOSS VERSUS REPHASING OF a,b giving a considerable Majorana correlation in Fig. 2(b). While− TOPOLOGICAL PROPERTIES the calculated system is small, we remark that this asymptotic In Fig. 2(b),thesolidbluelineshowsthenon-Markovian nonlocal correlation is not due to phonon-mediated long- dynamics of θ(t )forσ 0.6(correspondingto B 0.07), range interaction and is of topological origin, as we can show which decays to a substantial= value. Compared to⟨ the⟩= Marko- that the correlations decay with relative spacing, whereas the vian decoherence which eventually destroys Majorana modes edge-edge correlation is significant (see Appendix C). (solid black line), the long-lived and substantial Majorana correlation seen in the non-Markovian dynamics is quite re- V. CRITICAL MEMORY DEPTH markable, particularly given that Hp,s is near the topological phase boundary due to a significantly suppressed renormal- We find the edge dynamics can exhibit distinct relaxation ized pairing ! B !.Indeed,withoutthedissipationin behavior depending crucially on the memory depth, tunable Eq. (3), the dynamics⟨ ⟩≪ formally reduces to that of a coherent through the bandwidth σ of fermion-phonon coupling. Fig- quench in the pairing from ! to ! B .There,theMajorana ure 2(c) presents the non-Markovian dynamics of θ(t )for ⟨ ⟩ 245115-3 KAESTLE, SUN, HU, AND CARMELE PHYSICAL REVIEW B 102,245115(2020) various σ .Comparedtoσ 0.6[bluelineinFig.2(b)], an 1 initial decrease of σ results= in a steeper monotonic decay of U = 0.002 U = -0.002 θ(t )andasmallerasymptoticvalue[bluelineinFig.2(c)]. U = 0.000 U = -0.008 However, when σ decreases further, the monotonic relaxation transits into a nonmonotonic one: While the short-time 0.5 decoherence is accelerated, a buildup of edge correlation nonetheless occurs at some large times (purple line). Such a (t) buildup becomes stronger with decreasing σ,approachingan θ asymptotic value larger than the σ 0.6case(turquoiseand = 0 green lines). Strikingly, once σ surpasses a critical value, the 0 0.25 0.5 asymptotic Majorana correlation approaches θ(t ) 1(or- 100 Jt ange and dashed red lines). The inset in Fig. 2(c)∞summarizes→ the nonmonotonic variation of θ(t ): When σ decreases from FIG. 3. Non-Markovian Majorana dynamics for the renormal- ∞ alargevalue,θ(t )firstdecreasestoaminimumandthen ized Hamiltonian H in the presence of weak p-wave interaction, ∞ p,s increases toward unity. N 1 † † Hint U l −1 (cl cl 1/2)(cl 1cl 1 1/2). Calculations are per- Insights into the above intriguing phenomena can be ob- formed= at N = 4, J −! 0.01,+ µ+ −0, and σ 0.6. tained from the fact that reducing σ leads to an increased ! = = = = = lifetime of φ(τ )andthereforeincreasedmemory,atthecost of a smaller B exp[ φ(0)/2] [see Fig. 2(a)]. The for- Majorana correlation for subcritical memory depth (orange mer enhances⟨ the⟩= timescale− of the time-reversed evolution and red lines). An intuitive understanding can be obtained of Xa,b( τ )andhencetherephasingofpairing,whereasthe by noting that the attractive interaction is energetically fa- − latter further suppresses the bulk gap E! of Hp,s as well vorable for coherent formation of superconductive pairing, as weakening the memory strength. When σ is initially de- which provides a mechanism to counteract the aforemen- creased from 0.6, the latter effect dominates, aggravating tioned phonon-induced dephasing. This is consistent with the the decay. With further reduction of σ,however,theformer observation that for U > 0, θ(t )significantlydeclinesfrom rephasing effect grows, allowing phonons and fermions to the U 0caseatlongtimes(purpleline),asrepulsiveinter- synchronize and hence inducing backflow of parity informa- actions= energetically suppress pairing. tion. Consequently, a new dressed state manages to emerge, Summarizing, we have demonstrated memory-critical with buildup of Majorana correlation starting to dominate edge dynamics in a topological superconductor with non- over decoherence. In general, we estimate the recovery of Markovian interaction with phonons, resulting in a revival correlation can begin at times t >τM if E!τM ! 1issatisfied. of topological properties. We show this intriguing phe- Importantly, the existence of a critical σ indicates a criti- nomenon uniquely arises from the interplay between the cal memory depth, above which the system asymptotically phonon-renormalized topological Hamiltonian and the quan- approaches a new polaronic steady state in dynamical equilib- tum memory effect that simultaneously induces dephasing rium with phonons at T 4K,whichcanremarkablyexhibit and information backflow. Our analysis is based on the Ki- θ 1. The critical value= of σ in our case is between 0.21 and taev chain, but we expect the essential physics to occur for 0.20,≈ corresponding to B 0.01, but it is model specific. awideclassoftopologicalmaterialscoupledtoasuper- We note that the fermion-phonon⟨ ⟩= coupling bandwidth can be Ohmic reservoir. Currently, there are significant efforts in controlled, such as in solid-state setups with nanotechnologi- the condensed-matter context aimed at realizing Majorana cal design, e.g., alloys, impurities, and confinement potentials fermions based on the hybrid superconductor and semicon- [67–70]. ductor nanowires. The phonon-fermion coupling described here is relevant for InAs nanowires [29–34], where the acous- VI. CONCLUDING DISCUSSION tic phonons of InAs are in the range of typical nanowire length realization up to 100 nm and have super-Ohmic coupling as The central results of our work shown in Fig. 2 are investigated in our study. In a broader context, we anticipate found to be robust for initially nonideal Kitaev chains (see the fundamental non-Markovian feature of reservoir-induced Appendix D)andotherformsofsuper-Ohmiccoupling. recovery of topological properties may also be seen in cold- In the presence of perturbation caused by weak attrac- atom setups where phonons stem from the excitations of tive p-wave interactions, we find a speedup of the revival the superfluid reservoir coupled to ultracold quantum gases of Majorana correlation. Specifically, in Fig. 3,wecal- [71–73]orinsetupswhereacouplingtobosonsisinten- culate the non-Markovian evolution of θ(t )byincluding tionally induced as long as such coupling is structured and N 1 † † Hint U l −1 (cl cl 1/2)(cl 1cl 1 1/2) with interaction induces non-Markovian system-reservoir correlations. We fi- = = − + + − strength U 1inHp,s of Eq. (3)forσ 0.6. The inter- nally note our work is different from some recent work on action is|! treated|≪ exactly in our numerical solution,= which is topological systems which also considered non-Markovian mandatory since resorting to approximate descriptions such as system-bath interactions. In Refs. [74–76]animpurityora amean-fieldapproachwouldinevitablydestroyentanglement qubit is employed as a non-Markovian quantum probe of a in the system over time. Compared to the U 0case(blue topological reservoir, and Ref. [77]showsthatatopological line), we see that adding a weak attractive interaction= U < 0 phenomenon induced by a Markovian dissipation can persist not only shortens the time needed to reach the steady state, in non-Markovian regimes. In contrast, our study demon- but it can further enhance, depending on U ,theasymptotic strated that tailoring the coupling to a super-Ohmic reservoir | | 245115-4 MEMORY-CRITICAL DYNAMICAL BUILDUP OF … PHYSICAL REVIEW B 102,245115(2020) may suppress the loss of, and even fully restore, topologi- non-Markovian dynamics of Majorana correlation including cal properties, which opens an appealing prospect as to the Hshift.Inaddition,intheinsetinFig.4,thesteady-state explorations and control of memory-dependent topological correlations for various σ are compared for the cases includ- phenomena. ing the polaron shift (solid line) and disregarding it (dashed line). As can be seen from Fig. 4,includingthepolaronshift ACKNOWLEDGMENTS does not change the qualitative behavior and memory-critical buildup of the Majorana correlation. The polaron shift term O.K. and A.C. acknowledge support from the Deutsche is proportional to the width σ of the coupling. However, this Forschungsgemeinschaft (DFG) through SFB 910 project B1 merely affects the number of nonzero phonon modes taken (Project No. 163436311). Y.H. acknowledges the National into account for the energy renormalization, resulting in a Natural Science Foundation of China (Grant No. 11874038). plateau of the steady-state correlation at intermediate σ and slightly decreased steady-state values at large σ (see the inset APPENDIX A: POLARON SHIFT plot in Fig. 4). This is not to be confused with the large impact of σ on the correlation described in the main text, which The polaron transformation of the open-system Hamilto- originates from the strong dependence of the lifetime of the nian yields an additional term, phonon correlation φ(τ )onσ. 2 2 N 3 gk † Hshift d k c cl , (A1) = ω l k * l 1 + APPENDIX B: POLARON MASTER EQUATION % $= which is known as the polaron shift describing a polaron- For the description of the non-Markovian open-system induced energy renormalization. In the present study, we are dynamics, the polaron master equation is employed. It justified to neglect this polaron shift under proper conditions: is derived in second-order perturbation theory of the In the context of non-Markovian dynamics, it is standard polaron-transformed, dressed-state system-reservoir Hamilto- practice to neglect this polaron shift if the initial condition is nian. Phonon contributions are traced out in the process, while equilibrium,alongthelinesofLeggett(see,e.g.,Refs.[60,65] higher-order contributions of the fermion-phonon interactions and references therein). In our study, the initial state is as- by the phonon-renormalized Hamiltonian are still accounted sumed to be the ground state of a Kitaev chain, and therefore, for [53–58,60,61]. For the derivation of the master equation the polaron shift is ignored according to the convention. We in the polaron picture, the perturbative expansion is performed do not discuss a generic account of the polaron shift beyond with respect to the phonon-renormalized superconducting gap the aforementioned condition, which remains, to date, an open ! B ,with B 1. The parameter ! B is defined in the question in the context of the non-Markovian community and polaronic⟨ ⟩ frame⟨ ⟩≪ and addresses the strong⟨ fermion-phonon⟩ cor- is, of course, beyond the scope of our current work. relations in question. Nonetheless, to demonstrate that our qualitative results are The von Neumann equation ∂t ρ(t ) i/h¯[H,ρ(t )] is for- invariant against this energy renormalization, we provide cal- mally solved by integration and reinserted= into itself. First, it is culations in the presence of the polaron energy shift, which are assumed that the environment is large compared to the system treated numerically exactly. In Fig. 4,weshowtheresulting and initially in a thermal equilibrium state, known as the bath assumption. As a consequence, the reservoir is assumed to be a Gaussian bath with a defined temperature. Under these 1 conditions, the Born approximation then is applied, assuming aweaksystem-bathinteraction.Asaconsequenceofweak coupling between the system and reservoir, the latter is not 0.5 perturbed by excitation transfer from the system. Hence, it remains in its initial equilibrium state for all times, ρB(t )

(t) ρ (0), and the density matrix factorizes under these assump-≈ θ B 1 tions [60]: ρ(t τ ) ρS (t τ ) ρB(0), with indices S and 0 ) − ≈ − ⊗ ∞ B denoting the system and reservoir parts of the density ma- σ = 0.4 (t θ trix ρ(t ), respectively. Here, τ denotes past times that are

σ = 0.2 0.2 0.2 σ 0.8 taken into account for the calculation of the current state -0.5 within the scope of the second-order perturbative treatment. 0 0.5 1 In addition, the first Markovian approximation is applied, 100 Jt during which the history of the system part of the state is disregarded: ρS (t τ ) ρS (t ). It is assumed that the current FIG. 4. Non-Markovian dynamics of the polaronic Majorana − ≈ correlation θ(t )forvariousbandwidthsσ of the phonon coupling. state is mainly determined by the previous system state alone, Here, calculations include the polaron shift term Hshift. The corre- which is again valid if the influence from the reservoir is sponding steady state value θ(t ) is shown as a function of σ in the weak. However, the interaction between the system and the inset, comparing the cases including∞ the polaron shift (solid line) and reservoir itself maintains a history, as the influence of the disregarding it (dashed line). We have used N 4, J ! 0.01, phonons is taken into account on their own timescale. As a = = = 1 µ 0, fph 0.1, and phonon modes within k [0.0, 4.0] nm− are result, the interaction has a memory. Tracing out the phonon considered.= = ∈ degrees of freedom results in the non-Markovian Redfield

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iHp,sτ iHp,sτ master equation for the reduced system density matrix ρS [60], Xa,b( τ ) e− Xa,be .Weget,foraspecificmatrix element,− = ∂t ρS (t ) iTrB [Hp,s Hp,b Hp,I,ρS (t ) ρB] =− { + + ⊗ } † t m Xa,b( τ ) n m U (τ,0)Xa,b s s U (τ,0) n ⟨ | − | ⟩= ⟨ | | ⟩⟨ | | ⟩ dτTrB [Hp,I, [Hp,I( τ ),ρS (t ) ρB]] . (B1) s − 0 { − ⊗ } ${ } % † We define collective system and reservoir operators s U (τ,0) n m U (τ,0)Xa,b s = ⟨ | | ⟩⟨ | | ⟩ s :ρc (0) N 1 ${ } = − † † Xa(t ) ! (cl cl 1 cl 1cl ), (B2) Tr ρc( τ )Xa,b ,. /0 1 (B10) =− + + + l 1 = { − } = N$1 with n n1, n2,...,nN and site occupations nl 0, 1 . − | ⟩=| ⟩ ={ } † † Here, we introduced the conditional density matrix ρc( τ ), Xb(t ) ! (cl cl 1 cl 1cl ), (B3) = + − + − l 1 whose time evolution dynamics must be determined for = $ all possible initial conditions ρc(0) n m ,i.e.,with and =| ⟩⟨ | n ρc(0) m 1andallotherentriesbeingzero.Thedynam- † † ⟨ | | ⟩= 2[R(t ) R (t )] 2[R(t ) R (t )] ics of ρc(t )isprovidedby∂t ρc(t ) i[Hp,s,ρc(t )]. For the Ba(t ) e − e− − , (B4) = + numerical solution of the integro-differential=− polaron master 2[R(t ) R† (t )] 2[R(t ) R† (t )] equation, we employ two nested fourth-order Runge-Kutta Bb(t ) e − e− − , (B5) = − algorithms: (i) The first time integration is required to eval- † 3 † with R d k (gk/ωk )rk .TheFranck-Condonrenormal- uate the integral over all past times, which consists of the = iHp,sτ iHp,sτ ized, polaron-transformed interaction Hamiltonian then takes system correlators Xa,b( τ ) e− Xa,be and the time- the form " dependent phonon correlation− = function φ(τ ). (ii) In a second 1 1 time integration, the reduced system time evolution dynamics Hp,I(t ) Xa(t )[Ba(t ) Ba ] Xb(t )Bb(t ). (B6) = 2 −⟨ ⟩ + 2 are calculated using the previously determined integral up to As a next step, the integrand of Eq. (B1)iscalculated. the current time step. Assuming the phonon number nk is a Bose distribution, result- To investigate the robustness of Majorana edge states in ing in (2nk 1) coth [¯hωk/(2kBT )], and making use of the the Kitaev chain, we calculate the Majorana edge-edge cor- Baker-Campbell-Hausdorff+ = formula, the time-reversed bath relation. In the case of an ideal Kitaev chain it is given by † † correlations of the form Ba,bBa,b( τ ) , Ba,b( τ )Ba,b can i γLγR (t ) Tr ρ(t )(c1 c1 )(cN cN ) .Assumingeven ⟨ − ⟩ ⟨ − ⟩ − ⟨ ⟩ = { N + − } be evaluated. We define the phonon correlation function parity conditions, i.e., l 1 nl 2ν, ν N0 in the Jordan- Wigner phase factor, this yields= = ∈ 3 2 ! φ(τ ) d k 2gk (σ )/ωk = | | θ (t ) [ 1, n2,...,nN 1, 0 ρ(t ) 0, n2,...,nN 1, 1 % ⟨ ⟩ = ⟨ − | | − ⟩ n h¯ωk ${ } coth cos (ωkτ ) i sin(ωkτ ) (B7) × 2kBT − 0, n2,...,nN 1, 1 ρ(t ) 1, n2,...,nN 1, 0 , ' ( - +⟨ − | | − ⟩ and arrive at the final form of the bath correlations, 0, n2,...,nN 1, 0 ρ(t ) 1, n2,...,nN 1, 1 +⟨ − | | − ⟩ BaBa( τ ) 4exp[ φ(0)] cosh [φ(τ )], 1, n2,...,nN 1, 1 ρ(t ) 0, n2,...,nN 1, 0 ]. ⟨ − ⟩= − +⟨ − | | − ⟩ BbBb( τ ) 4exp[ φ(0)] sinh [φ(τ )], (B8) (B11) ⟨ − ⟩=− − with B B ( τ ) B ( τ )B .Insertingtheseex- a,b a,b a,b a,b ∗ APPENDIX C: TOPOLOGICAL CORRELATION pressions⟨ into− Eq.⟩ (=B1⟨), we− arrive⟩ at the polaron master equation Due to the numerically expensive calculation of the density t matrix and the finite memory kernel of the polaron mas- φ(0) ∂t ρS (t ) i[Hp,s,ρS (t )] e− dτ ter equation, the considered chain size is chosen at N =− − = %0 4sites.Here,wedemonstratethattheobservedMajorana ( cosh [φ(τ )] 1 [Xa, Xa( τ )ρS (t )] correlation at long times is genuinely of topological origin, × { − } − rather than caused by phonon-mediated long-range correla- sinh [φ(τ )][Xb, Xb( τ )ρS (t )] H.c.), (B9) − − + tions. To this end, we calculate the asymptotic correlation which, due to the limits of the applied approximations, is valid limt iTr[ρS (t )b1b2 j] between the Majorana operators →+∞{− } for the case of weak system-reservoir interactions [41,42]. b1 and b2 j for lattice sites j 2, 3, 4. Figure 5 shows the re- The initial separability of the system and reservoir is justified sulting correlation dynamics= for widths σ 0.6, 0.2 of the by the assumed initial state featuring a Majorana correlation. fermion-phonon coupling element. We see={ that the correlation} Afterwards, separability is ensured by the bath assumption decays with the separation d j 1andbecomesvanish- and Born approximation as long as we remain in the weak- ingly small for d 2butincreasestoattainasignificantvalue= − coupling regime B 1, where the polaron master equation at the right edge for= d 3. We see this phenomenon as a topo- holds. ⟨ ⟩≪ logical signature reflecting= the nonlocal Majorana correlation For the calculation of the full system-reservoir memory because if the beyond-nearest-neighbor correlation for sep- kernel, the time evolution of the system correlators Xa,b( τ ) aration d > 1originatesfromphonon-mediatedeffects,one is determined by the static part of the polaron Hamiltonian,− would expect a monotonous decay with the two-site distance.

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1 σ=0.6 1 σ = 0.6 σ = 0.3 =0.2 fL σ = 0.4 σ fR 1 0 1 2 site 3 4 0.5 0.5 correlatio n (t) θ

0 0 0 1 2 2 3 4 100 Jt site FIG. 6. Dynamics of the edge-edge correlation i b b (t ) − ⟨ 1 2N ⟩ = FIG. 5. The asymptotic correlation i b1b2 j (t ) iTr[ρS (t )b1b2N ]foranonidealKitaevchainwithJ !, µ 0.1J, − ⟨ ⟩ ∞ = − = = limt iTr[ρS (t )b1b2 j ] between Majorana operators b1 and N 4. Results for several coupling widths σ are shown. →+∞{− } = and b2 j for lattice sites j 2, 3, 4 in a Kitaev chain of length N 4, respectively calculated= at fermion-phonon coupling widths = σ 0.6, 0.2 . We take J ! 0.01, µ 0, fph 0.1. ideal case, the Majorana wave function fL/R,i extends into the ={ } = = = = bulk. But for very small deviations, it is still possible to ﬁnd suitable parameter regimes to ensure a small overlap between APPENDIX D: NONIDEAL KITAEV CHAIN Majorana modes for a four-site system. In Fig. 6,wepresent In the main text we showed our results with an initially calculations for the edge-edge correlation iTr[ρS (t )b1b2N ] ideal Kitaev chain, where the overlap between the left and for an initially nonideal initial Kitaev Hamiltonian− with J ! right Majorana modes is strictly zero even for a small system and µ 0.1J,andstill,weobserverevivalofthecorrelation= of four sites. If the initial Kitaev chain deviates away from the when σ=is tuned below a threshold value.

[1] N. Lindner, G. Refael, and V. Galitski, Nat. Phys. 7,490(2011). [14] J. W. McIver, B. Schulte, F.-U. Stein, T. Matsuyama, G. Jotzu, [2] T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, G. Meier, and A. Cavalleri, Nat. Phys. 16,38(2020). I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. [15] A. Elben, J. Yu, G. Zhu, M. Hafezi, F. Pollmann, P. Zoller, and Commun. 3,882(2012). B. Vermersch, Sci. Adv. 6,eaaz3666(2020). [3] G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, [16] M. S. Rudner and L. S. Levitov, Phys. Rev. Lett. 102,065703 D. Greif, and T. Esslinger, Nature (London) 515,237(2014). (2009). [4] D. Aasen, M. Hell, R. V. Mishmash, A. Higginbotham, J. [17] S. Diehl, E. Rico, M. A. Baranov, and P. Zoller, Nat. Phys. 7, Danon, M. Leijnse, T. S. Jespersen, J. A. Folk, C. M. 971 (2011). Marcus, K. Flensberg, and J. Alicea, Phys. Rev. X 6,031016 [18] D. Rainis and D. Loss, Phys. Rev. B 85,174533(2012). (2016). [19] M. J. Schmidt, D. Rainis, and D. Loss, Phys. Rev. B 86,085414 [5] Y. Hu, P. Zoller, and J. C. Budich, Phys. Rev. Lett. 117,126803 (2012). (2016). [20] M. Ippoliti, M. Rizzi, V. Giovannetti, and L. Mazza, Phys. Rev. [6] J. C. Budich, Y. Hu, and P. Zoller, Phys. Rev. Lett. 118,105302 A 93,062325(2016). (2017). [21] L. Mazza, M. Rizzi, M. D. Lukin, and J. I. Cirac, Phys. Rev. B [7] N. Fläschner, D. Vogel, M. Tarnowski, B. S. Rem, D.-S. 88,205142(2013). Lühmann, M. Heyl, J. C. Budich, L. Mathey, and K. Sengstock, [22] C.-E. Bardyn, M. A. Baranov, C. V. Kraus, E. Rico, A. Nat. Phys. 14,265(2018). Imamo˙ glu,˘ P. Zoller, and S. Diehl, New J. Phys. 15,085001 [8] M. Tarnowski, F. N. Ünal, N. Fläschner, B. S. Rem, A. Eckardt, (2013). K. Sengstock, and C. Weitenberg, Nat. Commun. 10,1728 [23] F. L. Pedrocchi and D. P. DiVincenzo, Phys. Rev. Lett. 115, (2019). 120402 (2015). [9] M. Heyl, Rep. Prog. Phys. 81,054001(2018). [24] Y. Hu, Z. Cai, M. A. Baranov, and P. Zoller, Phys. Rev. B 92, [10] W. Sun, C.-R. Yi, B.-Z. Wang, W.-W. Zhang, B. C. Sanders, 165118 (2015). X.-T. Xu, Z.-Y. Wang, J. Schmiedmayer, Y. Deng, X.-J. Liu, S. [25] J. C. Budich, P. Zoller, and S. Diehl, Phys. Rev. A 91,042117 Chen, and J.-W. Pan, Phys. Rev. Lett. 121,250403(2018). (2015). [11] B. Song, C. He, S. Niu, L. Zhang, Z. Ren, X.-J. Liu, and G.-B. [26] V. V. Albert, B. Bradlyn, M. Fraas, and L. Jiang, Phys. Rev. X Jo, Nat. Phys. 15,911(2019). 6,041031(2016). [12] L. Zhang, L. Zhang, S. Niu, and X.-J. Liu, Sci. Bull. 63,1385 [27] Z. Gong, S. Higashikawa, and M. Ueda, Phys. Rev. Lett. 118, (2018). 200401 (2017). [13] Z. Shi, P. W. Brouwer, K. Flensberg, L. I. Glazman, and F. von [28] S. Lieu, M. McGinley, and N. R. Cooper, Phys. Rev. Lett. 124, Oppen, Phys. Rev. B 101, 241414(R) (2020). 040401 (2020).

245115-7 KAESTLE, SUN, HU, AND CARMELE PHYSICAL REVIEW B 102,245115(2020)

[29] Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. 105, [54] S. Weiler, A. Ulhaq, S. M. Ulrich, D. Richter, M. Jetter, P. 177002 (2010). Michler, C. Roy, and S. Hughes, Phys. Rev. B 86,241304(R) [30] P. W. Brouwer, M. Duckheim, A. Romito, and F. von Oppen, (2012). Phys. Rev. Lett. 107,196804(2011). [55] R. Manson, K. Roy-Choudhury, and S. Hughes, Phys. Rev. B [31] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. 93,155423(2016). Bakkers, and L. P. Kouwenhoven, Science 336,1003(2012). [56] C. K. Lee, J. Moix, and J. Cao, J. Chem. Phys. 136,204120 [32] L. P. Rokhinson, X. Liu, and J. K. Furdyna, Nat. Phys. 8,795 (2012). (2012). [57] D. P. S. McCutcheon and A. Nazir, New J. Phys. 12,113042 [33] A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, and H. (2010). Shtrikman, Nat. Phys. 8,887(2012). [58] H.-T. Chang, P.-P. Zhang, and Y.-C. Cheng, J. Chem. Phys. 139, [34] S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon, J. Seo, 224112 (2013). A. H. MacDonald, B. A. Bernevig, and A. Yazdani, Science [59] The polaron master equation is based on the first Markovian 346,602(2014). approximation, disregarding the history of the system part of [35] H. O. H. Churchill, V. Fatemi, K. Grove-Rasmussen, M. T. the state, while the system-reservoir interaction maintains a Deng, P. Caroff, H. Q. Xu, and C. M. Marcus, Phys. Rev. B history, as phonon influences are taken into account on their 87, 241401(R) (2013). own timescale. The resulting master equation holds in the weak- [36] A. D. K. Finck, D. J. Van Harlingen, P. K. Mohseni, K. Jung, coupling regime. and X. Li, Phys. Rev. Lett. 110,126406(2013). [60] H. P. Breuer and F. Petruccione, The Theory of Open Quantum [37] V. Fatemi, S. Wu, Y. Cao, L. Bretheau, Q. D. Gibson, K. Systems (Oxford University Press, Oxford, 2002). Watanabe, T. Taniguchi, R. J. Cava, and P. Jarillo-Herrero, [61] A. Carmele and S. Reitzenstein, Nanophotonics 8,655 Science 362,926(2018). (2019). [38] D. Laroche, D. Bouman, D. J. van Woerkom, A. Proutski, C. [62] M. Strauß, A. Carmele, J. Schleibner, M. Hohn, C. Schneider, Murthy, D. I. Pikulin, C. Nayak, R. J. J. van Gulik, J. Nygrd, P. S. Höfling, J. Wolters, and S. Reitzenstein, Phys. Rev. Lett. 122, Krogstrup, L. P. Kouwenhoven, and A. Geresdi, Nat. Commun. 107401 (2019). 10,245(2019). [63] E. V. Denning, J. Iles-Smith, and J. Mork, Phys. Rev. B 100, [39] A. Fornieri, A. M. Whiticar, F. Setiawan, E. Portolés, A. C. C. 214306 (2019). Drachmann, A. Keselman, S. Gronin, C. Thomas, T. Wang, R. [64] E. V. Denning, J. Iles-Smith, N. Gregersen, and J. Mork, Opt. Kallaher, G. C. Gardner, E. Berg, M. J. Manfra, A. Stern, C. M. Mater. Express 10,222(2020). Marcus, and F. Nichele, Nature (London) 569,89(2019). [65] A. Caldeira and A. Leggett, Phys. A (Amsterdam, Neth.) 121, [40] H. Ren, F. Pientka, S. Hart, A. T. Pierce, M. Kosowsky, L. 587 (1983). Lunczer, R. Schlereth, B. Scharf, E. M. Hankiewicz, L. W. [66] We solve the integro-differential equation numerically via two Molenkamp, B. I. Halperin, and A. Yacoby, Nature (London) nested fourth-order Runge-Kutta algorithms: The first time in- 569,93(2019). tegration is required to evaluate the integral over all past times [41] H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Rev. Mod. τ. In a second time integration, the reduced system dynamics Phys. 88,021002(2016). are calculated using the previously determined integral up to [42] I. de Vega and D. Alonso, Rev. Mod. Phys. 89,015001(2017). the current time step. [43] A. Y. Kitaev, Phys. Usp. 44,131(2001). [67] W. Kim, J. Mater. Chem. C 3,10336(2015). [44] G. D. Mahan, Many-Particle Physics (Springer, Berlin, 2013). [68] S. Xiong, K. Sääskilahti, Y. A. Kosevich, H. Han, D. Donadio, [45] M. Kira and S. W. Koch, Semiconductor Quantum Optics and S. Volz, Phys. Rev. Lett. 117,025503(2016). (Cambridge University Press, Cambridge, 2011). [69] A. A. Balandin, J. Nanosci. Nanotechnol. 5,1015(2005). [46] V. May and O. Kühn, Charge and Energy Transfer Dynamics in [70] A. A. Balandin, E. P. Pokatilov, and D. L. Nika, J. Nanoelectron. Molecular Systems (Wiley, Hoboken, NJ, 2008). Optoelectron. 2,140(2007). [47] M. A. Stroscio and M. Dutta, Phonons in Nanostructures [71] L. Jiang, T. Kitagawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. (Cambridge University Press, Cambridge, 2001). Refael, J. I. Cirac, E. Demler, M. D. Lukin, and P. Zoller, Phys. [48] J. Förstner, C. Weber, J. Danckwerts, and A. Knorr, Phys. Rev. Rev. Lett. 106,220402(2011). Lett. 91,127401(2003). [72] S. Nascimbène, J. Phys. B 46,134005(2013). [49] B. Krummheuer, V. M. Axt, and T. Kuhn, Phys. Rev. B 65, [73] Y. Hu and M. A. Baranov, Phys. Rev. A 92,053615 195313 (2002). (2015). [50] A. Carmele, A. Knorr, and F. Milde, New J. Phys. 15,105024 [74] F. Grusdt, N. Y. Yao, D. Abanin, M. Fleischhauer, and E. (2013). Demler, Nat. Commun. 7,11994(2016). [51] N. Német, S. Parkins, A. Knorr, and A. Carmele, Phys. Rev. A [75] A. Camacho-Guardian, N. Goldman, P. Massignan, and G. M. 99,053809(2019). Bruun, Phys. Rev. B 99, 081105(R) (2019). [52] D. Reiter, T. Kuhn, and V. M. Axt, Adv. Phys.: X 4,1655478 [76] G. L. Giorgi, S. Longhi, A. Cabot, and R. Zambrini, Ann. Phys. (2019). (Berlin, Ger.) 531,1900307(2019). [53] I. Wilson-Rae and A. Imamoglu,˘ Phys. Rev. B 65,235311 [77] A. Ricottone, M. S. Rudner, and W. A. Coish, Phys. Rev. A 102, (2002). 012215 (2020).

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